# Binomial expansion

Binomial expression an expression (a + b)n which is the sum of two terms raised to the power n.

e.g. (x + 3)2

Binomial expansion (a + b)n expanded into a sum of terms

e.g. x+ 6x + 9

Binomial expansions get increasingly complex as the power increases:

The general formula for each term in the expansion is nCr an-r br .

In order to find the full binomial expansion of a binomial, you have to determine the coefficient nCr and the powers for each term. The powers for an and b are found as n − r and r respectively, as shown by the binomial expansion formula.

#### Binomial expansion formula

DB 1.9

The powers decrease by 1 for a and increase by 1 for b for each subsequent term.

The sum of the powers of each term will always = n.
There are two ways to find the coefficients: with Pascal’s triangle or the binomial coefficient function (nCr). You are expected to know both methods.

#### Pascal’s triangle

Pascal’s triangle is an easy way to find all the coefficients for your binomial expansion. It is particularly useful in cases where:

1.  the power is not too high (because you have to write it out manually)
2. you need to find all the terms in a binomial expansion

#### Binomial coefficient functions

The alternative is to calculate the individual coefficients using the nCr function on your calculator, or with the formula below.

(Note:In the 1st term of the expansion r = 0, in the 2nd term r = 1, . . .)

Expanding binomial expressions

1. Use the binomial expansion formula

2. Find coefficients using Pascal’s triangle for low powers or nCr on calculator for high powers

3. Put the terms and their coefficients together

4. Simplify using laws of exponents

Finding a specific term in a binomial expansion

Find the coefficient of x5 in the expansion of (2x − 5)8

1. Use the binomial expansion formula

(a + b)n =··· + nCan−r br +……

2. Determine r

Since a = 2x, to find x5 we need a5.

a5 = an−r = a8−r, so r = 3

3. Plug r into the general formula

nCan−r br8Ca8−3 b8Cab3

4. Replace a and b

8C(2x)(−5)3

5. Use nCr to calculate the value of the coefficient, nCr

8C3 = 8C3 = 56

6. Substitute and simplify

56 × 25(x5) × (−5)3 = −224000(x5)

⇒ coefficient of x5 is −224000

The IB use three different terms for these types of question which will effect the answer you should give.

Coefficient the number before the x value
Term the number and the x value
Constant term the number for which there is no x value (x0)